Before the match, the lights go dark and a group of 20 or more guys run to the ring to "fix" something. Within those guys is undertaker but you can't tell. All you know is that a group of people wearing all black runs to the ring. Within that time, Undertaker slips under the ring and waits for his song.
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When the lights go out, they go out for around 5 - 10 seconds, this is enough time for the Undertaker to escape.. How does he do it? He simply leaves the ring and hides under the ring.
When he appears during a match or speech from another superstar, he is already under the ring, when the lights go out he crawls out and climbs into the ring, making it look like the "Deadman" appeared out of nowhere.
I myself think the pyro and gimmicks for the WWE is incredible, hope this answers your question.
the under taker appears in the ring in the by being hidden the crowd and when they turn out the lights in the arena he comes out the crowd and enters the ring either from under the ring or from the side of it now you know or they would drop him from the ceiling on the fishing wire that is on his coat
the lights go out and he comes out from the ring or crowd the lights go out and he comes out from the ring or crowd